Loop Representation of Wigner's Little Groups

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Baskal S., Kim Y. S., Noz M. E.

SYMMETRY-BASEL, vol.9, no.7, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 9 Issue: 7
  • Publication Date: 2017
  • Doi Number: 10.3390/sym9070097
  • Journal Name: SYMMETRY-BASEL
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Keywords: Wigner's little groups, Lorentz group, unified picture of massive and massless particles, two-by-two representations, graphical approach to internal space-time symmetries, GAUGE TRANSFORMATIONS, MASSLESS PARTICLES, PHOTONS, INVARIANCE, ROTATIONS, SPIN
  • Middle East Technical University Affiliated: Yes

Abstract

Wigner's little groups are the subgroups of the Lorentz group whose transformations leave the momentum of a given particle invariant. They thus define the internal space-time symmetries of relativistic particles. These symmetries take different mathematical forms for massive and for massless particles. However, it is shown possible to construct one unified representation using a graphical description. This graphical approach allows us to describe vividly parity, time reversal, and charge conjugation of the internal symmetry groups. As for the language of group theory, the two-by-two representation is used throughout the paper. While this two-by-two representation is for spin-1/2 particles, it is shown possible to construct the representations for spin-0 particles, spin-1 particles, as well as for higher-spin particles, for both massive and massless cases. It is shown also that the four-by-four Dirac matrices constitute a two-by-two representation of Wigner's little group.